11-17-2025, 10:28 AM
Thread 5 — Non-Euclidean Geometry: Curved Space, Geodesics & Relativity
When Straight Lines Aren’t Straight — The Geometry of the Universe
Most people learn *Euclidean* geometry — the flat geometry of triangles, circles, and straight lines.
But our universe isn’t flat.
Space curves. Time bends. Light follows warped paths. Gravity isn’t a force — it’s geometry.
Welcome to *Non-Euclidean Geometry*: the mathematics of relativity, black holes, GPS satellites, and the shape of the cosmos.
1. Euclidean vs Non-Euclidean Geometry
Euclidean geometry assumes:
• Straight lines are truly straight
• Parallel lines never meet
• Triangles add to 180°
• Space is flat
But in non-Euclidean geometry:
• Straight lines bend
• Parallels can meet or diverge
• Triangle angles ≠ 180°
• Space itself is curved
This is the geometry Einstein placed at the heart of General Relativity.
2. Two Types of Curved Geometry
Hyperbolic Geometry (Negatively curved)
Examples:
• saddle surfaces
• inside of wormhole diagrams
• large-scale universe (current models suggest slight negative curvature)
Key features:
• Triangle angles sum to <180°
• Parallels diverge
• Space expands outward faster than Euclidean
Spherical Geometry (Positively curved)
Examples:
• Earth’s surface
• orbits around massive objects
• curved spacetime near stars
Features:
• Triangle angles sum to >180°
• Parallels meet
• “Straight lines” curve into great circles
3. Geodesics — The New “Straight Line”
In curved space, a straight line isn’t straight — it’s a geodesic.
A geodesic is:
• the shortest path between two points
• the path objects naturally follow without forces
Examples:
• Airplanes follow geodesics (great circles) on Earth
• Light follows geodesics when bending near stars
• Satellites orbit Earth along geodesics in curved spacetime
Gravity is literally just objects following curved geodesics.
4. Curved Space in Everyday Technology (GPS depends on it)
GPS satellites tick slower due to special relativity (speed).
They tick faster due to general relativity (weaker gravity).
The two effects don’t cancel — they leave a daily offset of 38 microseconds.
Without non-Euclidean corrections:
• GPS would drift by 10 km/day
• Google Maps would be useless
• Aircraft navigation would fail
Your phone works because the universe is curved.
5. Triangles in Curved Space — A Simple Demonstration
On a sphere (Earth):
Pick a point on the equator.
Walk 90° east.
Walk north until you reach the pole.
Walk down another great circle back to the equator.
Your triangle has:
• three 90° angles
• sum = 270°
• impossible in flat geometry
Curvature changes the rules.
6. Light Bending — Geometry, Not Forces
Einstein replaced Newton’s idea of gravitational *force* with:
\[
\text{Mass tells space how to curve.
Space tells objects how to move.}
\]
This is why:
• Light curves around stars
• Black holes trap light
• Gravitational lensing magnifies galaxies
• Time passes differently near massive objects
Everything is geometry.
7. Curvature and the Shape of the Universe
Three possibilities:
• Positive curvature (closed universe)
Finite but unbounded — like a 4D sphere.
• Zero curvature (flat universe)
Expands forever at constant geometry.
• Negative curvature (open universe)
Expands faster and faster.
Current cosmology suggests **near-flat but slightly negative** —
meaning the universe is almost Euclidean… but not quite.
8. Practice Problems — Think Like a Relativist
1. On a spherical world, can two “straight lines” meet twice?
2. Why do satellites in low orbit fall forward, not downward?
3. If triangle angles add to 190°, what type of curvature does the surface have?
4. Why does a laser beam curve slightly when fired near a massive planet?
5. Explain why “straight up” is not the same direction across the planet.
Written by Leejohnston & Liora — The Lumin Archive (Geometry & Space Division)
When Straight Lines Aren’t Straight — The Geometry of the Universe
Most people learn *Euclidean* geometry — the flat geometry of triangles, circles, and straight lines.
But our universe isn’t flat.
Space curves. Time bends. Light follows warped paths. Gravity isn’t a force — it’s geometry.
Welcome to *Non-Euclidean Geometry*: the mathematics of relativity, black holes, GPS satellites, and the shape of the cosmos.
1. Euclidean vs Non-Euclidean Geometry
Euclidean geometry assumes:
• Straight lines are truly straight
• Parallel lines never meet
• Triangles add to 180°
• Space is flat
But in non-Euclidean geometry:
• Straight lines bend
• Parallels can meet or diverge
• Triangle angles ≠ 180°
• Space itself is curved
This is the geometry Einstein placed at the heart of General Relativity.
2. Two Types of Curved Geometry
Hyperbolic Geometry (Negatively curved)
Examples:
• saddle surfaces
• inside of wormhole diagrams
• large-scale universe (current models suggest slight negative curvature)
Key features:
• Triangle angles sum to <180°
• Parallels diverge
• Space expands outward faster than Euclidean
Spherical Geometry (Positively curved)
Examples:
• Earth’s surface
• orbits around massive objects
• curved spacetime near stars
Features:
• Triangle angles sum to >180°
• Parallels meet
• “Straight lines” curve into great circles
3. Geodesics — The New “Straight Line”
In curved space, a straight line isn’t straight — it’s a geodesic.
A geodesic is:
• the shortest path between two points
• the path objects naturally follow without forces
Examples:
• Airplanes follow geodesics (great circles) on Earth
• Light follows geodesics when bending near stars
• Satellites orbit Earth along geodesics in curved spacetime
Gravity is literally just objects following curved geodesics.
4. Curved Space in Everyday Technology (GPS depends on it)
GPS satellites tick slower due to special relativity (speed).
They tick faster due to general relativity (weaker gravity).
The two effects don’t cancel — they leave a daily offset of 38 microseconds.
Without non-Euclidean corrections:
• GPS would drift by 10 km/day
• Google Maps would be useless
• Aircraft navigation would fail
Your phone works because the universe is curved.
5. Triangles in Curved Space — A Simple Demonstration
On a sphere (Earth):
Pick a point on the equator.
Walk 90° east.
Walk north until you reach the pole.
Walk down another great circle back to the equator.
Your triangle has:
• three 90° angles
• sum = 270°
• impossible in flat geometry
Curvature changes the rules.
6. Light Bending — Geometry, Not Forces
Einstein replaced Newton’s idea of gravitational *force* with:
\[
\text{Mass tells space how to curve.
Space tells objects how to move.}
\]
This is why:
• Light curves around stars
• Black holes trap light
• Gravitational lensing magnifies galaxies
• Time passes differently near massive objects
Everything is geometry.
7. Curvature and the Shape of the Universe
Three possibilities:
• Positive curvature (closed universe)
Finite but unbounded — like a 4D sphere.
• Zero curvature (flat universe)
Expands forever at constant geometry.
• Negative curvature (open universe)
Expands faster and faster.
Current cosmology suggests **near-flat but slightly negative** —
meaning the universe is almost Euclidean… but not quite.
8. Practice Problems — Think Like a Relativist
1. On a spherical world, can two “straight lines” meet twice?
2. Why do satellites in low orbit fall forward, not downward?
3. If triangle angles add to 190°, what type of curvature does the surface have?
4. Why does a laser beam curve slightly when fired near a massive planet?
5. Explain why “straight up” is not the same direction across the planet.
Written by Leejohnston & Liora — The Lumin Archive (Geometry & Space Division)